Engineering & Science
Lucas-Washburn Capillary Wicking Length & Assay Flow Time
Calculate horizontal capillary penetration distance L(t) = SQRT((γ * r * cos(θ) / (2 * μ)) * t) (mm) and lateral-flow diagnostic transit time t_flow (s) in porous paper strips and heat-pipe wicks.
Why lateral-flow diagnostic strips slow down quadratically as liquid advances!
When blood plasma or buffer wets a nitrocellulose rapid-test strip (or working fluid re-wets a sintered copper heat-pipe wick), Young-Laplace capillary suction ΔP = 2γ cos(θ) / r stays constant while Hagen-Poiseuille viscous drag grows linearly with wetted length L! Balancing those two forces yields the celebrated Lucas-Washburn (1921) square-root imbibition law: L(t) = SQRT((γ * r * cos(θ) / (2 * μ)) * t)!
Notice the design implication for microfluidic diagnostics: doubling test-line distance from 20 mm to 40 mm quadruples assay wait time! For aqueous buffer (γ = 0.072 N/m, μ = 0.001 Pa·s, contact angle θ = 35°) wicking through r = 8.0 μm nitrocellulose pores: after t = 10 s the front reaches L = 48.57 mm (reaching a 40 mm test line in just 6.78 s), whereas quadrupling time to t = 40 s only doubles distance to 97.14 mm!
| Imbibition Elapsed Time / Pore Regime | Pore Radius r & Contact θ | Wicked Distance L(t) (mm) | Square-Root Scaling |
|---|---|---|---|
| Rapid Strip Readout Window (t = 10 s) | r = 8.0 μm, θ = 35° | 48.57 mm | Baseline Front Position |
| Quadrupled Assay Time (t = 40 s) | r = 8.0 μm, θ = 35° | 97.14 mm (Exactly 2.0x!) | t -> 4t Doubles L(t) |
| Non-Wetting Hydrophobic Pore (θ >= 90°) | cos(θ) <= 0 | "Non-Wetting or Invalid Inputs" | Spontaneous Wicking Requires θ < 90° |
Lucas-Washburn Capillary Penetration Distance L(t) (mm)
Converts pore radius B3 from μm to m (* 1E-6) and wetting angle B4 to radians, returning wicked length in millimeters (* 1000).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 90, B5 <= 0, B6 < 0), "Non-Wetting or Invalid Inputs", ROUND(1000 * SQRT((B2 * (B3 * 0.000001) * COS(RADIANS(B4)) * B6) / (2 * B5)), 2))Transit Time Needed to Reach Target Test-Line Distance L_target (s)
Inverts Washburn’s equation t = 2 * μ * (L_mm / 1000)^2 / (γ * r * cos(θ)) for a 40 mm diagnostic readout line.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 90, B5 <= 0), "Non-Wetting or Invalid Inputs", ROUND((2 * B5 * ((40 / 1000) ^ 2)) / (B2 * (B3 * 0.000001) * COS(RADIANS(B4))), 2))Try the example
Use the original values or change them to test your understanding.